where \(f_{\beta \mu }\)  is the Fermi-Dirac function
\(f_{\beta \mu }(x)=\lbrace 1+e^{\beta (x-\mu )}\rbrace ^{-1}\) .
Here, the subscripts \(i\)  and \(j\)  denote the indices for an arbitrary
single-particle basis,
while \(\alpha \)  for the single-particle states to diagonalize \(H\)  (and \(H^{\prime }\) ),
\(\hat{H}| {\alpha } \rangle =\epsilon _\alpha | {\alpha } \rangle \) .
\(\hat{c}_i\)  (\(\hat{c}_i^\dagger \) ) is an annihilation (creation)
operator for a particle at the state \(| {i} \rangle \) .
Since the HF Hamiltonian \(\hat{H}[\rho ]\)  is a functional of
the one-body density, the states \(| {\alpha } \rangle \)  and energies \(\epsilon _\alpha \) 
depend on \(\rho _{ij}\) .
Thus, Eq. (REF ) should be iteratively calculated
until the self-consistency is achieved.
It is straightforward to extend the theory to
the HFB theory at finite temperature
[1]}, [2]}.
